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=KITI: Knowledge Interpretation Theory I= These notes are very much a work in progress and probably contain several mistakes. This theory is the foundation for Imaging Theories and the grounding for Knowledge Engineering. Please read the Copyright Left In Tact licence before you study this Theory. TEA: Time Expression Algebra TEA is an algebra of Points (instances) in Time. : \mathrm{From \; this \; point \; forward \; let: \; \; T} \equiv \mathrm{TEA} The complete definition of the algebraic structure of T is as follows. Note this is a Higher Order Algebra, i.e. operands and operators have parity. This is most evident in the action Tock where three operands transform into an operator. ---- : \begin{array}{rccc} \mathrm{Existence:} & & \vDash \!\, & 1 \\ \\ \mathrm{Substance:} & \sqrt{1} & \vDash \!\, & {-1} 1 \\ \\ \mathrm{Ignite:} & -1 & \leftrightarrow & i i \\ \\ \mathrm{Tick:} & i & \leftrightarrow & i ( i ( i ) i ) i \\ \\ \mathrm{Tock:} & ( i ( i ) i ) & \leftrightarrow & + \\ \\ \mathrm{Excite:} & i & \leftrightarrow & (( i i i ) ! ( i i )) \\ \end{array} ---- An instance of time T: : t \in \mathrm{T} is called a TEA TIME. A sequence of instances: : t_1, t_2, t_3, . \; . \; . is called a PAR TEA (sometimes mispelt PARTY) from the equality of TEA i.e. all Ts are equal. Unity Without Roots ---- : \begin{array}{rccc} \mathrm{Identity:} & & \vDash \!\, & \mathrm{I} \\ \\ \mathrm{+ \; beta:} & \mathrm{I} & \xrightarrow{\; \; + \beta \; \;} & \mathrm{I \; I} \\ \\ \mathrm{- \; beta:} & \mathrm{I \; I} & \xrightarrow{\; \; - \beta \; \;} & \mathrm{I} \\ \\ \mathrm{+ \; gamma:} & \mathrm{I \; I \; I \; I \; I} & \xrightarrow{\; \; + \gamma \; \;} & \mathrm{I \; + \; I} \\ \\ \mathrm{- \; gamma:} & \mathrm{+} & \xrightarrow{\; \; - \gamma \; \;} & \mathrm{I \; I \; I} \\ \\ \mathrm{+ \; tau:} & \mathrm{I} & \xrightarrow{\; \; + \tau \; \;} & \mathrm{(I \; ! \; I)} \\ \\ \mathrm{- \; tau:} & \mathrm{(I \; ! \; I)} & \xrightarrow{\; \; - \tau \; \;} & \mathrm{I} \\ \end{array} ---- SET: Space Exploration Transforms Everything is expected to fail. This is the type theory of T. A CUP processes an image returning a new image by applying one of the transformation in forward motion or reverse motion: : \begin{array}{rlcl} \mathrm{Let: \;\; } & Act & \equiv & \{ \mathrm{Tick^{\pm 1}}, \mathrm{Tock^{\pm 1}}, \mathrm{Excite^{\pm 1}} \} \\ \\ \mathrm{Let: \;\; } & S^0 & \equiv & T \\ \\ \mathrm{Let: \;\; } & S^1 & \equiv & T \times Act \times T \\ \\ \mathrm{Let: \;\;} & S^i & \equiv & T \times Act^i \times T , \;\;\;\; i \in \aleph_0 \\ \\ \mathrm{Let: \;\;} & S^{\aleph_0} & \equiv & T \times Act^{\aleph_0} \\ \\ \mathrm{Let: \;\;} & S & \equiv & \bigcup_{i \in \aleph_0} S^i \cup S^{\aleph_0} \\ \end{array} CUP: Complex Universal Processors This section needs more work. The idea is that a CUP is a path through a space. : \begin{array}{rlcl} \mathrm{Let: \;\;} & \Pi_i^0 & \equiv & S^0 \\ \\ \mathrm{Let: \;\;} & \Pi_i^1 & \equiv & T \rightarrow S^1 \\ \\ \mathrm{Let: \;\;} & \Pi_i^j & \equiv & N \rightarrow \Pi^j \\ \\ \Pi_i^j.t.T & : & T \end{array} The following is not correct yet but it shows where things are going. Omega should be the universal CUP. : \begin{array}{rlcl} \Omega & = & \sum_{\pi \in \aleph_1}^{e \in \aleph_1} { \Pi_{\pi}^e } \end{array} SIP: Simple Interpretation Process Interpretations of TEA create models. A SIP of TEA defines an interpretation that maps onto the complex plane. The prime standard interpretation I_0 is centred on \mathrm{ ( 0 + 0i ) } . Secondary standard interpretations I_z are centred on points off the origin. Non standard interpretations are non-linear with respect to I_0 . Elements of T are called Images. : \begin{array}{rcl} I_0 [ i ] & = & \mathrm{i} \\ \\ I_0 \; [ t_1 \; t_2 ] & = & I_0 \; [ t_1 ] \times I_0 \; [ t_2 ] \\ \\ I_0 \; [ t_1 \; + \; t_2 ] & = & I_0 \; [ t_1 ] + I_0 \; [ t_2 ] \\ \\ I_0 \; [ t_1 \; ! \; t_2 ] & = & I_0 \; [ t_1 ]^{ I_0 \; [ t_2 ] } \end{array} TEA POT: Time Expression Algebra Partially Observable Theory The TEA POT brews the TEA, enables a CUP to be filled and refilled. An observer of the CUP can then take SIPs in its own time. Example 1. Root : \begin{array}{lll} & i i \\ Eval \\ & I_0 ( i i ) & = -1 \end{array} Example 2. Tick : \begin{array}{lll} & i i \\ Tick \\ & i i ( i ( i ) i ) i \\ Eval \\ & I_0 [ i i ( i ( i ) i ) i ] & = -1 \end{array} \mathrm{Example \;\; 3.} \;\; Tick \; \; Tick^{-1} : \begin{array}{lll} & i i \\ Tick \\ & i i ( i ( i ) i ) i \\ Tick^{-1} \\ & i i \\ Eval \\ & I_0 [ i i ] & = -1 \end{array} \mathrm{Example \;\; 4.1.} \;\; Tick \; \; Tock : \begin{array}{lll} & i i \\ Tick \\ & i i ( i ( i ) i ) i \\ Tock \\ & i i + i \\ Eval \\ & I_0 [ i i + i ] & = -1 + i \end{array} \mathrm{Example \;\; 4.2.} \;\; Tick \; \; Tock : \begin{array}{lll} & i i \\ Tick \\ & i ( i ( i ) i ) i i \\ Tock \\ & i + i i \\ Eval \\ & I_0 [ i + i i ] & = i - 1 \end{array} ---- ---- =Licence= This work is licenced under the Copyright Left In Tact licence. Anyone may use this work in any way. ---- ----